A quadratic equation is an equation with one variable represented as a polynomial that can be written as ax² + bx + c = 0 where a, b, and c are constants. Quadratic equations are studied for the insight they can provide to properties such as peak and vertex of a parabola and of course, the number of solutions they have. In this article, we will explore how many solutions a quadratic equation can have, discuss various types of quadratic equations, whilst walking through the process of solving for the solutions.

The Basics of Quadratic Equations

A quadratic equation is an equation of the form ax² + bx + c = 0 that results in a parabolic graph. It can be reduced to a number of other forms such as using an absolute value, or using an exponential form. The most common form is the standard form which looks like ax² + bx + c = 0 where ‘a’ can have a value other than one, potentially resulting in positive and negative coefficients. We can further simplify this to determine the number of solutions there are to a given equation.

Types of Quadratic Equations

There are three types of quadratic equations. The first type is a perfect square trinomial, which has one solution. The second type is a difference of two squares, which also has one solution. The third type is a regular polynomial, which has zero, one, or two solutions depending on the value of the discriminant. The discriminant of a quadratic equation is the part of the formula that tells you how many solutions it has. It is calculated as b² – 4ac and is most commonly represented by the letter ‘D’.

Solving Quadratic Equations

To solve a quadratic equation, we first use the completing the square method, followed by the quadratic formula. The completing the square method involves taking the coefficient of the x squared term (a), multiplying it by 1 over 2 then adding the result to both sides of the equation. The next step is to complete the square, which means adding the square of the coefficient of the x term to both sides of the equation. The next step is to solve for x, which is done by subtracting c from both sides of the equation and then dividing both sides by `a`. The result is two values for x, or two solutions.

Zero Solutions

If the discriminant (D) is less than 0, then it means that the quadratic equation has no real solutions. This happens when both solutions are imaginary numbers and not real numbers. For example, if we have an equation like 4x² – 12x + 9 = 0 then we could calculate the discriminant as -12² – 4(4)(9) = -144 < 0. This means there are no real solutions to this equation.

One Solution

If the discriminant (D) is equal to 0, then it means that the quadratic equation has only one real solution. This happens when both solutions are identical and equal zero. In cases such as these, we can solve the equation by substituting ‘x’ with zero in both sides of the equation. For example if we have an equation like 4x² – 8x + 4 = 0 then we can calculate the discriminant to be zero, as -8² – 4(4)(4) = 0. This means that both solutions are equal zero, meaning there is only one real solution for this equation.

Two Solutions

If the discriminant (D) is greater than 0, then it means that the quadratic equation has two real solutions. This happens when both solutions are different and not equal zero. In cases such as these, we can use the quadratic formula to calculate both solutions altogether. For example if we have an equation like x² + 7x + 5 = 0 then we can calculate the discriminant to be greater than zero, as 72 – 4(1)(5) = 4 > 0. This means that both solutions are different, meaning there are two real solutions for this equation.

Imaginary Solutions

If the discriminant (D) is less than 0, then it means that the quadratic equation has two imaginary solutions. This happens when both solutions are imaginary numbers and not real numbers. In cases such as these, we use the quadratic formula to calculate both imaginary solutions using the imaginary number (i). For example if we have an equation like 4x² – 12x + 16 = 0 then we can calculate the discriminant to be less than zero, as -12² – 4(4)(16) = -144 < 0. This means that both solutions are imaginary numbers, meaning there are two imaginary solutions for this equation.

Three Solutions

If the discriminant (D) is equal to 0, then it means that the quadratic equation has three real solutions. This happens when all three solutions are identical and equal zero. In cases such as these, we can solve the equation by substituting ‘x’ with zero in all three sides of the equation. For example if we have an equation like x³ + 4x³ + 4 = 0 then we can calculate the discriminant to be zero, as 12 – 12 = 0. This means that all three solutions are equal zero, meaning there are three real solutions for this equation.

Complex Solutions

In cases where there are more than two solutions, it is possible to have complex solutions. These complex solutions involve adding or subtracting complex numbers to obtain the desired result. For example if we have an equation like 3x³ – 15x + 10 = 0 then it can have more than two solutions since 15² – 12(3)(10) = 40 > 0. A complex solution would involve adding or subtracting complex numbers to satisfy this equation.

Conclusion

To conclude, when solving a quadratic equation, we must first determine its type and then proceed with solving it according to the type. From a given formula, we calculate its discriminant (D) which helps us determine if the roots are real or imaginary, and whether it has one or two (or even three) solutions. By following these steps, it becomes easier to determine how many solutions a quadratic equation can have and solve for them accordingly.